Rigidity of maximal holomorphic representations of K\"ahler groups

TL;DR

This paper classifies maximal holomorphic representations of K"ahler groups into Hermitian Lie groups, showing they are mostly associated with ball quotients or diagonal embeddings, and provides a complete description of these representations.

Contribution

It offers a complete classification of maximal holomorphic representations of K"ahler groups, linking them to geometric structures like ball quotients and diagonal embeddings.

Findings

Maximal holomorphic representations correspond to ball quotients for higher dimensions.

Most representations of Riemann surface groups deform to holomorphic ones.

The paper characterizes the structure of maximal representations via Milnor--Wood inequalities.

Abstract

We investigate representations of K\"ahler groups $\Gamma = \pi_1(X)$ to a semisimple non-compact Hermitian Lie group $G$ that are deformable to a representation admitting an (anti)-holomorphic equivariant map. Such representations obey a Milnor--Wood inequality similar to those found by Burger--Iozzi and Koziarz--Maubon. Thanks to the study of the case of equality in Royden's version of the Ahlfors--Schwarz Lemma, we can completely describe the case of maximal holomorphic representations. If $\dim_{\C}X \geq 2$, these appear if and only if $X$ is a ball quotient, and essentially reduce to the diagonal embedding $\Gamma < \SU(n,1) \to \SU(nq,q) \hookrightarrow \SU(p,q)$. If $X$ is a Riemann surface, most representations are deformable to a holomorphic one. In that case, we give a complete classification of the maximal holomorphic representations, that thus appear as preferred elements of the respective maximal connected components.